Estimating a Dynamic Firm Location Model with
Agglomeration Externalities∗
Jeff Brinkman
Carnegie Mellon University
Daniele Coen-Pirani
University of Pittsburgh
Holger Sieg
University of Pennsylvania
Preliminary and Incomplete: Do not Cite or Quote
July 8, 2010
∗ We
would like to thank Dennis Epple, Vernon Henderson, Thomas Holmes, and Matt
Kahn and seminar participants at Carnegie Mellon University for comments and suggestions.
Abstract
We develop a new dynamic general equilibrium model of firm location choice that
can explain the observed sorting of firms by productivity and is consistent with the
observed entry, exit, and relocation decisions of firms within an urban economy. We
discuss existence of equilibrium of and characterize the stationary distribution of
firms in each location. The parameters of the model can be estimated using a nested
fixed point algorithm. We implement the estimator using data collect by Dunn and
Bradstreet for the Pittsburgh metropolitan area. The data suggest that firms located
in the city are older and larger than firms located outside the urban core. As a
consequence they use more land and labor in the production process. However, they
face higher rental rates for land and office space. As a consequence they operate with
a higher employee per land ratio. We find that our model explains these observed
features of the data well. Finally, we consider the impact of different relocation
policies that provide targeted subsidies to new start-ups and superstar firms.
1
Introduction
Cities and metropolitan areas play in an important role in the economy since economic
proximity makes for more efficient production and trade.1 These efficiency gains
typically arise due to agglomeration externalities which arise because of synergies
between firms in the same industry. These efficiency can be due to lower transaction
costs or sharing of a common labor pool in an uncertain industry (Marshall, 1890).
Alternatively, the may arise due to synergies between different industries (Jacobs,
1969). Firms that operate in locations with high externalities, therefore, have a
competitive advantage over firms that are located in less efficient locations.2 Since
firms will bid for the right to locate in areas witch high agglomeration externalities,
these locations have higher land values and higher wages than locations that are less
efficient.
3
As a consequence, firms with different productivity levels will sort in
equilibrium with high productivity firms locating in areas with high agglomeration
effects and high rents. Low productivity firms are forced to exit the economy or
operate in cheaper locations.
As the productivity of a firm changes over time, a firm’s demand for land and labor
changes as well. Moreover, these productivity shocks create incentives to relocate
within the city to exploit a better match with the agglomeration externalities. A firm
that may have initially located in the suburbs may find it in its interest to move to
a more densely populated central business district in order to grow and capture the
full benefits of a persistent positive productivity shock. Similarly, a firm that has
1
The idea of geographic returns to scale was first introduced by von Thünen (1825).
Krugman (1991) provides theoretical foundations for a two-location model of agglomeration.
Ellison and Glaeser (1997) argued that agglomeration externalities are important to understand
geographic concentration of manufacturing in the U.S. The literature of agglomeration theory is
reviewed in Fujita and Thisse (2002) and Duranton and Puga (2004).
3
Anas and Kim (1996) and Lucas and Rossi-Hansberg (2002) have developed equilibrium models
of mono- and poly-centric urban land use with endogenous congestion and job agglomeration. RossiHansberg (2004) studies optimal land use policies in a similar framework.
2
1
experienced a persistent negative shock may find it in its interest to downsize and
move to the urban fringe where land and labor is cheaper than in the city.4 The first
objective of this paper is then to develop a new dynamic general equilibrium model
of firm location choice that can explain the observed sorting of firms by productivity
and is consistent with the observed entry, exit, and relocation decisions of firms in an
urban economy.
We consider an urban economy with two distinct locations. We can interpret the
two locations as the Central Business District and the rest of the metropolitan area.5
Our model assumes that these locations differ in their agglomeration externalities
which increase with employment density.
6
Firms are heterogeneous in their produc-
tivities. Agglomeration externalities enter into the production function of firms and
thus increase the productivity of firms. To model the evolution of firm productivity,
our model builds Hopenhayn (1992)’s analysis of firm dynamics in equilibrium. Firms
enter our urban economy with an initial productivity and must pay an entry cost.
Productivity then evolves according to stochastic first order Markov process. Each
period firms compete in the product market, must pay a fixed costs of operating,
4
There is some evidence that shows that agglomeration effects are important to understand firm
dynamics. Henderson, Kunkoro, and Turner (1995) show that agglomeration effects for mature
industries are related to Marshall scale economies, while newer industries benefit from diversity
akin to Jacobs economies. This work is important because it points to agglomeration as part of
a dynamic process. Other research has continued to study the relevance of agglomeration in firm
life-cycle dynamics. Duranton and Puga (2001) study the the effect of agglomeration externalities in
innovation and the development of production processes, while Dumais, Ellison, and Glaeser (2002)
examine the effect of firm dynamics (entry, exit, expansion, and contraction) on the concentration
of economic activity.
5
An alternative interpretation of our model is that we focus on two cities with different externalities.
6
Deckle and Eaton (1999) find that geographic scale of agglomeration is mostly at the national
level, while the financial sector is concentrated in specific metropolitan areas. Other work finds
that agglomeration can occur on a much more local scale. In particular, Rosenthal and Strange
(2001, 2003) establish the level and type of agglomeration at different geographic scales, and also
the measure the attenuation of these externalities within metropolitan areas. Holmes and Stevens
(2002) find evidence of differences in plant scale in areas of high concentration, suggesting production externalities act on individual establishments.A review of empirical evidence of agglomeration
economies is found in (Rosenthal and Strange, 2004).
2
and realize a profit. Entry, exit and relocations are dynamic and based on expectations of future productivity shocks. Using recursive methods, we can characterize the
optimal decision rules for firms in each location as well as potential entrants. Low
productivity firms exit from the economy while high productivity firms continue to
operate. Relocation choices are driven by the interaction of agglomeration effects
and firm productivity shocks. We assume that large firms with higher productivity
shocks prefer locations with high agglomeration externalities relative to smaller, less
productive firms. As a consequence, a high productivity firm that is located outside
the central business district may have strong incentives to relocate to the city center.7
We define the stationary equilibrium of our model. We show that that the model
can generate multiple equilibria. The first type of equilibrium has entry in both
locations. The second type of equilibrium only has entry in one of the two locations.
In both types of equilibria, there is a stationary distribution of firms by age and
productivity for each location. If we shut down future shocks, we can then prove that
firms follow simple cut-off rules in equilibrium. Moreover, we can provide sufficient
conditions that guarantee that a unique equilibrium with entry in both locations
exists. Finally, we can analytically characterize the unique stationary distribution of
firms in each location. For general versions of the model, we provide an algorithm
that can be used to compute equilibria and check for local uniqueness.8
The second objective of the paper is to estimate the parameters of our model
and determine whether our model can explain the observed sorting of firms in one
metropolitan area. We focus on equilibria with entry in both locations since this is
7
There are some similarities with the literature that studies the sorting pattern of household in
urban areas which starts with the classic papers by Alonso (1964), Mills (1967), and Muth (1969).
8
Related to our research is also work by Melitz (2003) who studies the impact of trade on intraindustry reallocations. Rossi-Hansberg and Wright (2007) examine the relationship of establishment
scale and entry and exit dynamics. Finally, Combes, Duranton, Gobillon, Puga, and Roux (2010)
distinguish between selection effects and productivity externalities by estimating productivity distributions across cities.
3
a common feature of the data. The parameters of the model can then be estimated
using nested fixed point algorithm. The inner loop computes the equilibrium for each
parameter value, while the outer loop searches over feasible parameter values. Our
simulated methods of moment estimator matches the observed distribution of firms
by age, size and land use by location to the one predicted by our model.
We implement the estimator using data collect by Dunn and Bradstreet for the
Pittsburgh metropolitan area. Since regional cities often act as a hub for services for
a larger region, we focus on locational choices within that sector. The data suggest
that firms located in the city are older and larger than firms located in the rest of the
metro area. As a consequence they use more land and labor in the production process.
However, they face higher rental rates for land and office space. As a consequence
they operate with a higher employee per land ratio. We find that our model explains
these observed features of the data reasonably well. The parameter estimates have
the expected sign and are highly significant. Using the estimate model, we perform
a number of policy experiments. ...
The rest of the paper is organized as follows. Section 2 provides some stylized
facts that characterize firm location choices with U.S. cities. Section develops our
stochastic, dynamic equilibrium model and discusses its properties. Section 4 discusses estimation of our model. Section 5 provides the data set used in our application. Section 6 discusses the empirical results and discusses the policy experiments.
Section 7 offers some conclusions that can be drawn from the analysis.
2
Firm Location Choices in U.S. Cities
The purpose of this paper is to provide a quantitative analysis of locational decisions
of firms within a given metropolitan area using a dynamic general equilibrium frame-
4
work. To get some quantitative insights into firms sorting behavior, we collected
Census data for a number of metro areas. We are mostly interested in characterizing the sorting of establishments by age, employment, and facility size. We define
a business district within a metropolitan area as those zip codes within a city that
have a high density of firms signifying local agglomeration. To make this concept
operational, we use an employment density of at least 10,000 employees per square
mile. These locations need not be contiguous, as some metropolitan areas exhibit
multiple dense business districts.
We pay special attention to service industries, given that there is strong evidence
that large U.S. cities have undergone a transformation over the past decades moving
from centers of individual sectors toward becoming hubs of headquarters or business
services. Duranton and Puga (2005), for example, show evidence that cities have
become more functionally specialized, with larger cities, in particular, emerging as
centers for headquarters and business services. They posit that this change is primarily related to industrial structure, and a decrease in remote management costs in
particular. In addition, Davis and Henderson (2008) provide further evidence that
services and headquarters are indeed more concentrated in large cities relative to the
entire economy, and that headquarter concentration is linked to availability of diverse
services.
We exclude wholesale and retail businesses from our analysis of services. Holmes
and others have shown that retail locational decisions are primarily driven by proximity to costumers (Hotelling, 1929).9 For similar reasons, we also do not consider
businesses in the entertainment sector. Finally, we omit businesses related to agriculture, forestry, mining and fishing for fairly obvious reasons. We thus define the
service sector as consisting of businesses that operate in information, finance, real
9
See Bresnahan and Reiss (1991) and Holmes (2010) for some structural empirical studies of
retail location.
5
estate, professional services, management, administrative support, education, health
care and related services. We find evidence that the concentration of services is also
prevalent at a much more local level, with the service industry choosing to locate in
dense business districts.
Table 1: Concentration of employment in dense business districts
Total
% Emp. Avg.
Avg.
% Service
Emp.
CBD
Emp.
E mp. EstabMSA
CBD
lishments
MSA
Atlanta
1,344,400 17.03%
17.13
29.25
46.29%
Boston
2,259,424 23.52%
18.22
39.01
43.90%
3,598,916 14.69%
16.73
24.47
44.05%
Chicago
Columbus
768,812
8.23%
19.02
23.73
43.91%
518,501
3.62%
17.49
26.95
40.79%
Hartford
Houston
2,007,199 14.28%
17.43
28.47
44.80%
Jacksonville
516,274
4.71%
15.53
25.38
43.72%
5,231,962 18.63%
15.68
19.39
45.34%
Los Angeles
Philadelphia
2,118,054 9.27%
16.56
27.66
44.55%
1,616,714 4.01%
18.56
27.78
48.39%
Phoenix
979,022
16.04%
16.23
40.04
40.55%
Pittsburgh
Salt Lake
493,325
10.76%
15.69
21.08
46.68%
San Antonio
682,312
3.89%
17.32
20.49
43.65%
1,439,565 12.45%
15.08
20.33
43.59%
Seattle
St Louis
1,237,993 6.79%
17.09
42.57
41.71%
2,234,618 13.59%
16.05
21.68
50.96%
Wash. DC
MSA
% Service
Establishments
CBD
63.31%
59.90%
66.50%
58.64%
61.41%
65.51%
66.28%
52.39%
55.74%
71.01%
60.90%
58.90%
56.59%
58.97%
52.43%
60.05%
Table 1 shows the concentration of employment in dense business districts for a
sample of U.S. cities. First, we report statistics using all firms that are located in the
metro area. we find that there is a significant amount of heterogeneity among the
cities in our sample. There are some cities such as Phoenix and Hartford which have
few districts with a high density of firms. Most larger cities in the US such as New
York, Los Angeles, Chicago, Boston, Washington, Philadelphia, and Houston have
a significant fraction of firms located in high density central business districts. This
finding is also true for a variety of mid sized cities such as Pittsburgh and Seattle.
6
Focusing only on firms in the service sector a much clearer picture emerges. The
service industries are more concentrated and tend to be located in the dense business
districts. Moreover, there is much more commonality among metropolitan areas.
Firms located in the CBD are larger than the MSA average which indicates that they
have higher levels of productivity. This findings is common among all cities in our
sample.
3
Theory
We develop a stochastic dynamic model with two locations which differ in their agglomeration externalities.
3.1
A Dynamic Model of Firm Location
We consider a model with two locations, denoted by j = 1, 2. There is a continuum
of firms that produce a single output good and compete in the product market.10
In each period a firm chooses to stay where it is, relocate to the other location, or
shutdown. Firm are heterogeneous and productivity evolves according to a stochastic
law of motion.
Assumption 1 In each time period, a firm is subject to a productivity shock,ϕ. The
productivity shock evolves over time according to a Markov process with a conditional
distribution F (ϕ0 |ϕ).
In our parametrized model, we assume that the logarithm of the productivity
shock follows an AR(1) process, i.e. log(ϕ)0 = ρ log(ϕ) + ε0 , where ρ is the correlation
10
If there is only one location and agglomeration effects are irrelevant, our model is identical to
the one studied in Hopenhayn (1992) and Hopenhayn and Rogerson (1993).
7
coefficient and ε is a normally distributed random variable with mean µ and variance
σ2 .
Firm produce a single output good using labor and land as input factors. The
technology that is available to the firms in the economy satisfies the following assumption.
Assumption 2 The production function of a firm in location j can then be written
as:
q = f (ϕ, n, l; ej )
(1)
where q is output, n is labor, l is land, and ej is the agglomeration externality in
location j. The production function satisfies standard regularity conditions.
Rosenthal and Strange (2003) suggest that the externality acts as a multiplier on
the production function. We use a standard Cobb-Douglas function with parameters
α and γ in our computational model, q = ϕ ej nα (l − ¯l)γ . Note that ¯l is a minimum
amount of land required for production. Since rj ¯lj can also be interpreted as fixed
costs, this specification implies that fixed cost vary by location.
The agglomeration effects arise due to a high concentration of firms operating
at the same location. Following Lucas and Rossi-Hansberg (2002), we adopt the
following specification.
Assumption 3 The agglomeration externality can be written as
ej = Θ(Lj , Nj , Sj )
(2)
where Nj and Lj are aggregate measures of labor and land respectively, and Sj is a
measure of the mass of firms in location j.
8
In our computational model we assume that
ej =
θ
Nj
Lj − Sj ¯l
(3)
If θ > 0, the externality is, therefore, an increasing function of a measure of concentration of economic activity in a location j. This measure is represented by the ratio
of the total number of workers and the amount of land used in production over and
above the minimum land requirement.
The urban economy is part of a larger economic system which determines output
prices and wages.
Assumption 4 Output prices, p, and wages, w, are constant and determined exogenously.
Rental prices, rj , however, are equilibrium outcomes. The supply of land is determined by an inverse land supply function in each location.
Assumption 5 The inverse land supply function is given by:
rj = rj (Lj ), j = 1, 2
(4)
The inverse supply function is increasing in the amount of land denoted by Lj .
In the computational analysis, We adopt an iso-elastic functional form: rj =
Aj Lδj , j = 1, 2, where Aj and δ are parameters. Our model thus captures the fundamental trade-off faced by all firms in the urban economy. The benefits of the
agglomeration externality are at least partially offset by higher rents.
9
We can break down the decision problem of firms into a static and a dynamic
problem. First, consider the static part of the decision problem that a firm has to
solve each period. This problem arises because firm compete in the product market
each period.
Assumption 6 The product market is competitive and firms behave as price takers. Firms make decisions on land and labor usage after they have observed their
productivity shock, ϕ, for that period.
Let πj denote a firm’s one period profit in location j. The static profit maximization problem can be written as:
{n, l} = arg max πj (n, l; ϕ) ,
(5)
{n,l}
where the profit function is given by:
πj (ϕ) ≡ πj (n, l; ϕ) = p f (ϕ, n, l; ej ) − wn − rj l − cf .
The parameter cf denotes a fixed cost of operation independent of location. Solving
this problem we obtain the demand for inputs as a function of ϕ, denoted by nj (ϕ)
and lj (ϕ). as well as an indirect profit function.11
Let µj denote the measure of firms located in j, then the mass of firms located
located in j, Sj , is given by the following expression:
Z
Sj =
µj (dϕ)
11
(6)
Note that the sub-index j summarizes the dependence of the profit and input demand functions
on location j’s rent and externality.
10
Given the static choices for land and labor use for each firm, we can also calculate
the aggregate level of land and labor:
Z
Lj =
lj (ϕ)µj (dϕ),
(7)
nj (ϕ)µj (dϕ)
(8)
Z
Nj =
After choosing labor and land inputs, each firm faces the (dynamic) decision of
whether to stay in its current location, move to the other location, or shut down. The
following Bellman equations formalize the decision problem of a firm that begins the
period in location j with a productivity shock ϕ:
Z
Z
0
0
0
0
V1 (ϕ) = π1 (ϕ) + β max 0, V1 (ϕ ) F (dϕ |ϕ), V2 (ϕ ) F (dϕ |ϕ) − cr
(9)
Z
Z
0
0
0
0
V2 (ϕ) = π2 (ϕ) + β max 0, V2 (ϕ ) F (dϕ |ϕ), V1 (ϕ ) F (dϕ |ϕ) − cr
where β is the discount factor, cr is the cost of relocating from one location to another.
Solving the dynamic decision problem above implies decision rules of the following
form for firms currently in location j:



0 if f irm exits in next period


xj (ϕ) =
1 if f irm chooses location 1 in next period



 2 if f irm chooses location 2 in next period
(10)
To close the model, we need to specify the process of entry.
Assumption 7 Firms can enter into both locations. All prospective entrants are ex11
ante identical. Upon entering a new firm incurs cost cej and draws a productivity
shock ϕ from a distribution ν(ϕ).
Note that we allow the entry cost to vary by location. In our parametrized model
2
the entrant distribution is assumed to be log-normal with parameters µent and σent
.
These assumptions guarantee that the expected discounted profits of a prospective
firm are always less or equal than the entry cost:
Z
cej ≥
Vj (ϕ) ν (dϕ) , j = 1, 2
(11)
If there is positive entry of firms, then this condition holds with equality.
We are now in a position to define the equilibrium to our economy.
Definition 1 A stationary equilibrium for this economy consists of rents in each
location, rj∗ (j = 1,2), masses of entrants in each locations, Mj∗ (j = 1,2), a stationary
distributions of firms in the two locations, µ∗j (ϕ), externalities in each location, e∗j ,
land demand functions , lj∗ (ϕ), labor demand functions , n∗j (ϕ), value functions for
each location, Vj∗ (ϕ), and decision rules for location in the next period, x∗j (ϕ), such
that:
1. The decision rules (10) for a firm’s location are optimal, in the sense that they
maximize the right-hand side of equations (9).
2. The decision rules for labor and land inputs solve the firm’s static problem in
(5).
3. The free entry conditions (11) are satisfied in each location.
4. The market for land clears in each location consistent with equation ( 4).
12
5. The mass of firms in each location is given by equation (6).
6. The externalities are consistent with (2)
7. The distributions of firms µ∗j are stationary in each location and consistent with
firms’ decision rules.
3.2
Existence and Computation of Equilibrium
It should be clear from the discussion so far that there is no hope to solve for equilibrium analytically. Hence we need to rely on numerical techniques to compute
equilibria which is also quite challenging. An equilibrium to our model is characterized by vector of equilibrium values for rents, mass of entrants, and externalities in
each location (r1 , r2 , M1 , M2 , e1 , e2 ). Finding a equilibrium for this model is equivalent to the problem of finding the root of a nonlinear system of equations with size
equations. For any vector (r1 , r2 , M1 , M2 , e1 , e2 ), we can
1. solve the firms’ static profit maximization problem and obtain land demand,
labor demand, and the indirect profit functions for each location;
2. solve the dynamic programming problem in equations (9) and obtain the optimal
decision rules;
3. chose an initial mass of entrants in each location and simulate the economy
forward until the distribution of firms, µj , converges to a stationary distribution;
4. calculate the aggregates land, labor demands and supplies the economy;
5. check whether market clearing conditions and the equations that define the
mass of firms in each location are satisfied.
13
If they are not, we update the vector of scalars and repeat the process until all of
the conditions for equilibrium are satisfied.
The task of computing an equilibrium can be simplified by exploiting some properties the parametrization used in our computational model. The static first order
condition that determines that ratio of land and labor inputs is given by:
n
α rj
=
¯
γw
l−l
(12)
Notice that the ratio in this equation is the same for all firms in the same location j.
Aggregating over all firms in such location, we obtain that:
α rj
Nj
=
γw
Lj − Sj ¯l
(13)
Thus, equation (13) then implies that we obtain an expression linking the externality,
ej in each location to that location’s rent, rj . We can, therefore, solve the Bellman
equations without knowing the aggregate levels of land and labor. As a consequence
we characterize equilibrium rent values based on the free entry conditions.
To see how this works, we adopt the following simplified notation for the expected
values functions:
Z
EV1 (r1 , r2 ) =
V1 (ϕ)dν(ϕ)
(14)
Z
EV2 (r1 , r2 ) =
V2 (ϕ)dν(ϕ)
The entry condition for location one then defines a function r1 = Γ1 (r2 ), i.e. for given
r2 , Γ1 (r2 ) is the value of r1 such that EV1 (r1 , r2 ) = ce . Similarly, we can define a
14
function r1 = Γ2 (r2 ) for location two. These two functions then effectively define the
set of rent pairs {r1 , r2 }, such that the two free entry conditions are satisfied with
equality.
Figure 1: The Γj (r2 )-Locus
Figure 1 shows an example of these functions for a calibrated example. In our
example Γ1 (r2 ) and Γ2 (r2 ) intersect three times. As a consequence, there are three
possible candidate values for equilibria with entry in both locations. There is the
15
trivial solution in which r1 = r2 and then there are two solutions in which r1 6= r2 .12
Next let as assume that the elasticity of land supply, δ, is the same in both
locations. Define the ratio of entrants in the two locations as m =
M1
M2
and the
distribution of firms standardized by the mass of entrants in locations 2 as,
µ̂j =
µj
.
M2
The standardized stationary distributions satisfy,
µ̂1 (ϕ0 ) =
R
F (ϕ0 |ϕ) 1 {x1 (ϕ) = 1} µ̂1 (dϕ) +
R
F (ϕ0 |ϕ) 1 {x2 (ϕ) = 1} µ̂2 (dϕ) + mν(ϕ0 )
µ̂2 (ϕ0 ) =
R
F (ϕ0 |ϕ) 1 {x1 (ϕ) = 2} µ̂1 (dϕ) +
R
F (ϕ0 |ϕ) 1 {x2 (ϕ) = 2} µ̂2 (dϕ) + ν(ϕ0 )
where 1{xj (ϕ) = j} is an indicator function equal to 1 if xj equals j and 0 otherwise.
Given m forward iteration on these two equations yields the equilibrium standardized
stationary distributions µ
bj , j = 1, 2.
To find the equilibrium m, substitute the aggregate demands for land in the two
locations into the inverse land supply functions and take their ratios. We then obtain:
R
δ
l1 (ϕ)µ̂1 (dϕ)
r1
A1
R
=
.
r2
A2
l2 (ϕ)µ̂2 (dϕ)
12
(15)
In addition to equilibria with entry in both locations, it is also possible to have equilibria in
which entry only occurs in one of the two locations. We provide an example in Appendix C.
16
Let us define the right hand side of this equation as a function of r1 , r2 , and m, and
define the function rm (r2 ; m) as follows:
LR (rm (r2 ; m), r2 , m) =
A1
A2
(16)
Hence, r1 = rm (r2 ; m) is the value of r1 that clears the relative land markets give r2
and m.
We can thus conclude that all rent pairs {r1 , r2 } that are consistent with entry
in both locations are characterized by the intersection of the two functions Γj (r2 ).
In addition, we have characterized the set of rent pairs consistent with market clearing condition,rm (r2 ; m), corresponding to different values of m. By analyzing these
functions all together, we can completely characterize the set of triplets {r1 , r2 , m}
consistent with equilibrium in the economy. Figure 2 thus illustrates the equilibria
that can arise in our model.
JEFF: we want to focus on the equilibria with entry in both locations in the Figure
above.
Finally, the mass of entrants in location 2, M2 , is determined by the market
clearing condition for land:
r2
A2
1δ
Z
= M2
l2 (ϕ)µ̂2 (dϕ),
Note that M2 can be solved for analytically.
17
Figure 2:
18
3.3
Analytical Properties of Equilibrium
To get some additional insights into the properties of our model it is useful to simplify
the structure of the model and shut down the future productivity shocks. We can
then characterize the equilibrium of the model in closed-form. Let us impose the
following additional assumptions.13
Assumption 8
1. The shock is drawn upon entry once and for all from a uniform distribution in
[0, 1]:
ν (ϕ) = 1 for ϕ ∈ [0, 1] .
2. There are no fixed cost of operation: cf = 0.
3. Importance of externality: θ = 1 − α > γ
Let ξ < 1 denotes the exogenous exit probability (NOTE WE NEED TO INCLUDE THIS IN THE EARLIER PART OF THE MODEL). Let 1 denote the high
rent location and 2 the low rent one (1=city, 2=suburb). We show how to construct
a unique equilibrium in which r1 > r2 and firms move from location 2 to location 1,
but not vice versa. Firms who enter in location 1 stay there all the time or exit.
First note that under assumptions 2 and 3 above the profit functions can be
written as:
πj (ϕ) = rj ∆ϕη − l , j = 1, 2,
(17)
where ∆ > 0 and η > 1 are known functions of the parameters of the model (see the
appendix for details). Consider location in the city. We have the following result.
13
The model cannot be entirely solved in closed form because the equilibrium r2 has to satisfy a
highly non-linear equation. Sufficient conditions on the model’s parameters for r2 to exist and be
unique are imposed instead. Conditional on r2 , everything else can be solved for analytically.
19
Proposition 1 If r1 > r2 ,
a) Then firms in location 1 follow a simple cut-off rule. Firms below a threshold ϕl
exit while firms above the threshold stay in location 1 forever. The cut-off is defined
as:
ϕl =
l
∆
η1
.
b) Then firms in location 2 follow a simple cut-off rule. Firms below the threshold ϕl
exit, firms with shocks between ϕl and ϕh stay in location 2, and firms with shocks
larger than ϕh move to location 1. The cut-off ϕh is defined as:
ϕh =
l
cr (1 − βξ)
+
∆ ∆ (r1 − r2 )
η1
.
(18)
Proof:
a) Note that static firm profits are monotonically increasing in ϕ. Define ϕl such that
π1 (ϕl ) = 0. Then firms with ϕ < ϕl exit immediately. It is straight forward to show
that
V1 (ϕl ) = π1 (ϕl ) + βξ max {0, V1 (ϕl ) , V2 (ϕl ) − cr } = π1 (ϕl ) = 0
where the second equality follows from the fact that the productivity cut-off for
switching to location 2 is less then cut-off for exit if r1 > r2 as assumed. Firms with
ϕ > ϕl stay in location 1 as long as they survive the exogenous destruction shock ξ.
Their payoffs are:
V1 (ϕ) =
π1 (ϕ)
>0
1 − βξ
20
b) Next consider the decision rule of firms located in the suburb. Firms with ϕ < ϕl
exit immediately:14
V2 (ϕl ) = 0
Firms with shocks in (ϕl , ϕh ) stay in 2 forever (as long as they survive the exogenous
destruction shock). Firms with high shock move to 1. The indifference condition for
staying vs moving is:
π2 (ϕh )
= π2 (ϕh ) + βξ (V1 (ϕh ) − cr ) .
1 − βξ
(19)
This equation that defines the cut-off value ϕh . The lemma then follows from the
result that benefits of switching to location 1 monotonically increase with ϕ. Q.E.D.
Next we consider the free entry conditions and show that these conditions determine the rents in both locations. We have the following result:
Proposition 2 There is at most one set of rental rates (r1 , r2 ) that are consistent
with the entry in both locations. Conditions on the parameter values guarantee existence of (r1 , r2 ).
Proof (of uniqueness):
First consider the free entry condition in location 1 which is given by
Z
V1 (ϕ) ν (ϕ) dϕ = ce .
14
Note that equation (17) implies that zl does not depend on the location.
21
Substituting in our optimal decision rule and simplifying we obtain the equilibrium
rent in location 1:
r1 =
ce (1 − βξ) (η + 1)
.
∆ 1 − βξϕη+1
− l (1 − βξϕl ) (η + 1)
l
Free entry in location 2 requires:
Z
V2 (ϕ) ν (ϕ) dϕ = ce .
Replacing the value function in location 2 and taking into account the definition of
ϕh in (18) this equation simplifies to:
ηϕh 1+η − (1 + η) ϕηh − K = 0,
(20)
where K represents a non-positive combination of the parameters and is defined in
the appendix. The lhs of this equation is positive when K = 0 and has a negative
first derivative. Thus, if a solution for ϕh exists it must be unique. In turn, ϕh is
monotonically related to r2 by equation (18):
r2 = r1 −
cr (1 − βξ)
.
∆ϕηh − l
Thus, if the solution ϕh to equation (20) is unique, the equilibrium value of r2 is
also unique. The appendix provides sufficient conditions on the parameters for this
solution to exists. Q.E.D.
Next we characterize the equilibrium distribution of firms in each location.
22
Proposition 3 For each value of M2 , there exists unique stationary equilibrium distributions of firms in each location.
Proof:
Without loss of generality, let us normalize the model so that entry in location 2 is
always equal to M2 = 1. This implies a specific choice of A2 . Given this the mass of
firms in location 2 is µ̂2 (ϕ):
µ̂2 (ϕ) =









zl if ϕ < zl
1
1−ξ
(zh − zl ) if ϕ ∈ [zl , zh ] .
1 − zh if ϕ > zh
Note that firms in location 2 with ϕ < zl exit and there is a measure zl of them.
Firms with ϕ > zh move to 1, and there is a measure 1 − zh of them. Firms in the
middle group ϕ ∈ [zl , zh ] remain in 2 forever subject to surviving the death shock ξ.
Let m denote entry in location 1. The mass of firms in location 1 is:
µ̂1 (ϕ) =









m zl if ϕ < zl
m
(zh − zl ) if
1−ξ
(ξ+m)
(1 − zh )
1−ξ
ϕ ∈ [zl , zh ] .
if ϕ > zh
Firms in the first group exit immediately. Firms in the middle group stay in 1 forever.
Firms with ϕ > zh come from 2 sources: 1. firms who entered in 1 and stayed
there forever subject to death shock m (1 − zh ) / (1 − ξ) plus firms who entered in
location 2 last period, survived the shock and moved to 1 where they remain forever:
ξ (1 − zh ) / (1 − ξ) . Q.E.D.
Finally, we have the following result:
23
Proposition 4 There is at most one value of m such that the relative demand for
land equals the relative supply of land. Under conditions on the parameters, m is
shown to exist.
Proof:
Given the equilibrium distributions, we can solve for equilibrium value for entry,
denoted by m. Note that given the assumptions the demand for labor is:
lj (ϕ) = l +
α η
w
1
1−α
ϕ 1−α−γ rj1−α−γ .
The equilibrium value of m is such that it solves the relative land equilibrium condition
which can be written as
Z
A2 r 1
l1 (ϕ) µ̂1 (dϕ) =
A1 r 2
Z
l2 (ϕ) µ̂2 (dϕ)
(21)
where the right hand side does not depend on m. The left-hand side depends linearly
in m through the mass µ̂1 (ϕ) in an increasing way. This means that if m exists it is
unique.
For m → ∞ the left hand side of (21) goes to infinity. For m → 0 the left hand
side is strictly positive. To show that it is less than the right hand side you need
to have A1 sufficiently small. Note that we can always choose A1 small because this
is the first time this parameters shows up. Thus, there exists a unique value of m.
Q.E.D.
In what follows we present the equilibrium of the model in a numerical example.
Result 1 Consider the following parameter values: β = 0.5, α = 0.65, θ = 0.35,
ξ = 0.9, l = 0.01, γ = 0.01, η = 2.94, w = 1, ∆ = 0.0367, A1 = 0.5, A2 = 1.0,
24
ce = 0.1, cr = 0.01, δ = 1. Then, the unique quilibrium of the model is characterized
by the following: ϕl = 0.64, ϕh = 0.69, r1 = 37.18, r2 = 34.98, m = 0.21.
The analysis of this section shows that there exists a unique (up to scale) equilibrium with entry in both locations. Our analysis in the previous section reinforces the
notion that equilibria with entry in both locations are often locally unique. These
results suggest that we can design a full solution estimation algorithm for the parameters of the model that is based on equilibrium with entry in both locations.
4
Estimation
Let θ denote the parameter vector of the structural model to be estimated. The
equilibrium of our dynamic equilibrium model defines a nonlinear mapping from the
parameter θ to the distribution of observed equilibrium outcomes. This mapping is
implicitly defined by the equilibrium of the model analyzed in the previous section.
Since the econometrician does not necessarily observe all relevant variables, the empirical strategy relies on latent-variable simulation methods to estimate the parameters
of the model. The structural model is assumed to be true in the sense that there is a
particular value θ0 of the structural model and a realization of the exogenous variables
such that the observed data will correspond to the simulated data. The estimation
strategy relies on the idea that the structural model should be able to replicate the
empirical regularities observed in the data.
More precisely, the observed data are completely described by the joint empirical
distribution function of age, facility size, and employment conditional on location
choice. The basic idea to match these distributions with those generated by our
model. We illustrate this procedure, by considering the age distribution of firm in
each location. We can calculate the mass of establishments of each age recursively.
25
First we compute the equilibrium which gives us the entrant distribution, ν(ϕ); location decision rules, x1 (ϕ) and x2 (ϕ); conditional distribution of shocks, F (ϕ0 |ϕ); and
masses of entrants, M1 and M2 . We denote the distribution of establishment shocks
for each age t, and location j, as aj,t (ϕ). Then with the following initial conditions,
a1,1 (ϕ) = M1 ν(ϕ)
(22)
a2,1 (ϕ) = M2 ν(ϕ)
(23)
along with the following recursive equations,
0
Z
[a1,t−1 (ϕ)1{x1 (ϕ) = 1} + a2,t−1 (ϕ)1{x2 (ϕ) = 1}] F (ϕ0 |ϕ)dϕ (24)
Z
[a2,t−1 (ϕ)1{x2 (ϕ) = 2} + a1,t−1 (ϕ)1{x1 (ϕ) = 2}] F (ϕ0 |ϕ)dϕ (25)
a1,t (ϕ ) =
0
a2,t (ϕ ) =
we can calculate the complete distribution of shocks by age. The total mass of
establishments at each age t and location j, is then given by,
Z
agej,t =
aj,t (ϕ)dϕ
(26)
We then We can then forward simulate this process until we obtain convergence to a
obtain a stationary age distribution.
26
Instead of focusing on the joint empirical distribution of the observed variables,
one can restrict the attention to a number of select moments which one can try to
match along the lines suggested by Hansen (1982). We then match selected observed
moments with their simulated counterparts generated by the structural model.15
The moments we use are based on the joint distribution of age and employment,
and the distribution of facility size. These are calculated separately by location. In
addition, we use the total percentage of firms in the city relative to the entire county.
In practice, these moments are constructed by placing establishments into categories,
(e.g. 5 to 8 employees, 11 to 20 years old, located in the city). This is analogous
to creating a histogram of the distribution. The moments then are calculated as the
percentage of firms in a given category relative to the number of establishments in
the entire county.16
Combine all moments used in the estimation procedure into one vector mN and
denote with mS (θ) their simulate counterparts where S denotes the number of simulations. The orthogonality conditions are then given by
gN,S (θ) = mN − mS (θ)
(27)
Following Hansen (1982), θ can be estimated using the following moments estimator:
θN = arg min gS,N (θ)0 AN gS,N (θ)
θ∈Θ
15
(28)
Alternatively, one could estimate an auxiliary model using semi-nonparametric estimation as
developed by Gallant and Nychka (1987) and then match the scoring functions of the auxiliary and
the structural model (Gallant and Tauchen, 1989). This approach is more appealing for a time series
application since there are much more potential moments to chose.
16
The complete set of moments is described in Appendix B, along with the computational equivalents.
27
for some positive semi-definite matrix AN which converges in probability to A0 . Using standard asymptotic arguments, it follows that the estimator θN is a consistent
estimator of θ0 and that:
d
0
0
N 1/2 (θN − θ0 ) → N (0, (Ã0 D0 )−1 Ã0 V Ã0 (Ã0 D0 )−1 )
(29)
0
where Ã0 = D0 A0 , D0 = E [∂m(θ) / ∂θ0 ] and V is asymptotic covariance matrix of
the vector of sample moments. In addition, the most efficient estimator is obtained
by setting AN = VN−1 . In this case:
d
0
N 1/2 (θN − θ0 ) → N (0, (D0 V −1 D0 )−1 )
(30)
Furthermore, standard J-statistics can be used to do hypothesis and specification
tests.17
5
Data
Our empirical application focuses on firm location choices in the City of Pittsburgh
and Allegheny County.18 To illustrate the spatial distribution of economic activity in
Allegheny County, Figure 3 plots the employment concentration in the county using
data from the U.S. Census. Figure 3 shows that over 20 percent of employment is
concentrated in three zip codes in the center of Pittsburgh.19 These zip codes include
17
Strictly speaking, one would need to correct for the sampling error induced into the estimation
procedure by the simulations. However, if the number of simulations is large, these errors will be
negligible. (For a more careful discussion see Gourieroux and Monfort (1993).)
18
Appendix A of the paper compares to Pittsburgh to other metropolitan areas and finds that
similar firm location patterns can be found in other cities in the US.
19
The zip codes are 15222, 15219, and 15213.
28
Figure 3: Allegheny County and the City of Pittsburgh
29
the downtown business district and the Oakland neighborhood of Pittsburgh which
are the two significant dense commercial areas of Pittsburgh. The second location or
the suburbs include the rest of Allegheny County.
We analyze the employment and land use characteristics for different industries in
the Pittsburgh area. Table 2 reports the total employment, the average employment
and the land use per employee for firms in the city and the suburbs for the service
industries that are the focus of the empirical analysis.
NAICS
Table 2: Employment and facility size by
Total
% emp Avg.
Emp.
CBD
Emp.
Suburbs
Information
Finance
Real Estate
Professional Services
Management
Administrative Support
Education
Health Care
16,975
42,960
18,459
64,076
2,062
41,830
52,995
115,048
25.15%
53.51%
17.97%
32.85%
11.30%
14.97%
42.69%
18.12%
13.52
8.55
7.51
6.99
19.46
11.01
30.46
16.53
industry
Avg.
Emp.
CBD
31.16
55.66
12.43
13.29
14.56
20.67
205.66
24.01
sq. ft.
per employee
Suburbs
336.88
318.28
743.36
334.83
272.88
240.89
316.70
293.39
sq. ft.
per employee
CBD
214.44
193.59
1190.21
309.60
360.52
352.14
121.27
291.46
JEFF: we want to add other statistics such as revenue per employee or salaries per
employee here if we have that data.
We find that the service industries are much more heavily concentrated in the
CDB. The average employment size of establishments is larger in the central business
district reflecting an increase in productivity, and the facility space per employee is
lower in the CBD. The high concentration of economic activity in central business
districts is driven by the fact that regional cities act as a hub for services for a
larger region. The service industry concentrates in the city to reap the benefits of
agglomeration externalities available at an urban district scale.
For the remainder of the empirical analysis we will, therefore, focus on locational
30
Figure 4: Age, Facility Size, and Employment Distributions
31
choices in the services sectors which we define as consisting of all sectors in the lower
part of Table 2. To provide a more detail analysis of spatial activity of firms, we
obtained firm level data from Dun and Bradstreet’s Million Dollar Database20 in
2008 for all of Allegheny county. The database provides detailed information on
establishments and the coverage is near universal (compared to Census counts of
establishments in the county.) Of particular relevance to the current research, the
database provides data on location, facility size, total employment, industry, and
year established. While most of the data is complete, the year established field
(used to determine age of establishments) is only available for 52.5 percent of the
observations. However, there is little evidence that the missing data field is strongly
or systematically correlated with other observable data, and therefore entries missing
the ’year established’ field are dropped for analysis.21
Table 3 provides some descriptive statistics for firms located in the city and the
suburbs for these industries using our Dun and Bradstreet sample. Note that 13.43
percent of all our firms are located in the city and the rest in the suburbs. However,
the three zip codes that comprise the city only account for a small fraction of all the
land in Allegheny county.
percentile
10th
25th
50th
75th
90th
age
6
12
20
34
53
Table 3: Percentiles of data by location
City
Suburbs
employment facility size age employment facility size
1
890
4
1
780
2
1400
10
1
1000
5
2500
18
3
1700
14
4700
28
7
3000
40
11100
41
20
5800
A few clean patterns arise form the analysis of Table 3. Firms in the city are
20
Information on Dun and Bradstreet data is available on-line at http://www.dnbmdd.com/
In the appendix we provide details of the comparison between the total data set and the data
set with ’year established’ data based on observed data.
21
32
older, employ more workers and operate in larger facilities. This is especially true for
the right tail of the age distribution.
We also consider the dynamics of firm exist, entry, and relocations. U.S. Census
data for Allegheny County reports the number of new establishment and the number
of firm death per period. We collected this data for the period from 1999 through
2003. We find that the economy has been in steady state for a number of years. Take,
for example, the year 2003. The Census reports 2,871 establishments births and 2,989
deaths. Unfortunately, we do not have these data by firm sector and thus do not use
these statistics in estimation.
6
Empirical Results
6.1
Preliminary Parameter Estimates
To facilitate the estimation of the model use several parameters estimates from previous studies. Estimates about labor and land shares are reported in Deckle and Eaton
(1993), Adsera (2000), and Caselli and Coleman (2001). We set α, the labor share of
to be equal to 0.65. We set γ, the facility share, equal to 0.06. Moreover we set the
discount factor equal to 0.95. These are all relative standard choices. In addition,
we fix two other parameters. First, we set the relocation cost parameter cr equal
to 20,000. In the absence of reliable data on relocations it is hard to identify this
parameter. Second, we set the facility supply elasticity, denoted by δ, equal to one.
Again it is hard to estimate the supply elasticity given the data available to us.
The remaining parameters of the model are the estimated using the GMM estimator discussed above. Table 4 reports the parameter estimates and the estimated
standard.
33
Table 4: Parameter Estimates
Description
Parameter Point Estimate
production land share
γ
0.0883
externality
θ
0.0936
¯
minimum land
610
l
shock correlation parameter
ρ
0.97755
shock mean parameter
µ
0.24
shock variance parameter
σ
0.1356
entrant (log) mean
µent
11.5
entrant (log) variance
σent
0.36
fixed cost
cf
88,000
entry cost - location 1
ce,1
7.78E+05
entry cost - location 2
ce,2
7.81E+05
A1
land supply ratio
2.535
A2
Standard Errors
0.00024
0.0030
133.40
0.00094
0.0112
0.00107
0.00286
0.00617
3,495.08
3.37E+04
4.48E+04
0.918
We find that the land share parameter is approximate 0.088 which is less than
the parameter estimate of the agglomeration externality which is equal to 0.0936.
We have seen before that the restriction that θ > γ is often necessary to get an
equilibrium sorting pattern in which high productivity firms prefer locations with
high agglomeration externalities. The fixed costs of operation are $ 88,000, or three
times the costs of relocating to a different community. Entry costs differ by location
and vary between $778,000 and $781,000. It is only slightly more expensive to enter
into the suburbs. The minimum land requirement is approximately 610 square foot
with an estimate standard error of 130. The productivity shocks are highly correlated
across time. The point estimate of 0.977 is consistent with previous estimates in the
literature and points to a high a degree of persistence.
To obtain additional insights into the sorting process of firms we consider the
distribution of employment by age in both the city and suburbs. Table 5 reports the
distribution of firms by age and employment size for our sample and the one predicted
by our model. The data suggest that firms located in the city are older and larger
than firms located in the rest of the metro area. Moreover, they firms in the city
34
Table 5: Age-Employment distributions of establishments by location, as percentage
of entire county (computational moments in parenthesis)
City
Age
1 to 10
11 to 20 21 to 30
> 30
1 to 3
1.48 %
1.71 %
0.96 %
1.17 %
(2.00 %) (0.80 %) (0.48 %) (1.84 %)
Employment 4 to 8
0.59 %
1.10 %
0.56 %
0.93 %
(0.91 %) (0.57 %) (0.38 %) (1.63 %)
9 to 16
0.33 %
0.56 %
0.48 %
0.64 %
(0.35 %) (0.29 %) (0.23 %) (1.12 %)
> 16
0.41 %
0.64 %
0.63 %
1.24 %
(0.23 %) (0.37 %) (0.44 %) (2.53 %)
Suburbs
1 to 10
1 to 3
16.25 %
(17.99 %)
Employment 4 to 8
3.59 %
(8.18 %)
9 to 16
1.44 %
(3.18 %)
> 16
1.44 %
(1.93 %)
35
Age
11 to 20 21 to 30
16.00 %
9.48 %
(7.22 %) (4.29 %)
5.89 %
4.64 %
(5.08 %) (3.29 %)
2.69 %
1.81 %
(2.61 %) (1.86 %)
2.84 %
2.04 %
(2.36 %) (1.97 %)
> 30
8.17 %
(9.09 %)
4.44 %
(7.33 %)
2.31 %
(4.37 %)
3.50 %
(5.05 %)
have more employees holding age constant. Overall, we find that our model fits this
feature of the data reasonably well. The main drawback of the model is that it has
problem matching the age distribution of small firms that are located in the suburbs.
The vast majority of these firms have only one to three employees. A large number
of the small firms have been in business for a long time.
Table 6: Age-Facility Size establishment distributions by location as percentage of
county total, (computational moments in parenthesis
City
Age
1 to 10
11 to 20 21 to 30
> 30
1 to 1250
0.82 %
0.73 %
0.44 %
0.65 %
(1.58 %) (0.58 %) (0.35 %) (1.30 %)
Facility (sq ft) 1251 to 2150
0.64 %
1.18 %
0.59 %
0.88 %
(0.74 %) (0.40 %) (0.25 %) (0.69 %)
2151 to 3850
0.75 %
1.13 %
0.70 %
0.82 %
(0.56 %) (0.36 %) (0.25 %) (1.08 %)
> 3850
0.60 %
0.98 %
0.91 %
1.63 %
(0.62 %) (0.69 %) (0.69 %) (3.73 %)
Suburbs
1 to 10
1 to 1250
9.57 %
(13.51 %)
Facility (sq ft) 1251 to 2150
7.08 %
(6.22 %)
2151 to 3850
3.81 %
(5.80 %)
> 3850
2.30 %
(5.76 %)
Age
11 to 20 21 to 30
> 30
9.76 %
5.51 %
4.20 %
(4.93 %) (2.89 %) (6.08 %)
7.56 %
4.86 %
4.74 %
(3.26 %) (2.00 %) (4.15 %)
6.01 %
4.14 %
4.18 %
(3.66 %) (2.38 %) (5.30 %)
4.09 %
3.45 %
5.13 %
(5.43 %) (4.15 %) (10.14 %)
Table 6 focuses on the age-facility size establishment distributions. In our model,
employment and facility size are highly correlated since they both nonlinear functions
of the underlying productivity shock. A similar high correlation can be observed in
the data. As a consequence, Table 6 reinforces our previous findings about model fit.
Firms in the city face higher rental rates for land and office space. As a consequence
36
they operate with a higher employee per land ratio. Overall, the model fits this
feature of the data reasonably well.
7
Conclusions
37
References
Adsera, A. (2000). Sectoral Spillovers and the Price of Land: a Cost Analysis. Regional Science and
Urban Economics, 30.
Alonso, W. (1964). Location and Land Use: Towards a General Theory of Land Rent. Cambridge
University Press. Cambridge.
Anas, A. and Kim, I. (1996). General Equilibrium Models of Polycentric Urban Land Use with
Endogenous Congestion and Job Agglomeration. Journal of Urban Economics, 40 (2).
Bresnahan, T. F. and Reiss, P. C. (1991). Entry and Competition in Concentrated Markets. The
Journal of Political Economy, 99 (5).
Caselli, F. and Coleman, W. J. (2001). The U.S. Structural Transformation and Regional Convergence: A Reinterpretation. The Journal of Political Economy, 109.
Combes, P. P., Duranton, G., Gobillon, L., Puga, D., and Roux, S. (2010). The productivity
advantages of large markets: Distinguishing agglomeration from firm selection. Working
Paper.
Davis, J. C. and Henderson, J. V. (2008). The Agglomeration of Headquarters. Regional Science
and Urban Economics, 38 (5).
Deckle, R. and Eaton, J. (1993). Agglomeration and Land Rents: Evidence from the Prefectures.
The Journal of Urban Economics, 46.
Dumais, G., Ellison, G., and Glaeser, E. (2002). Geographic Concentration as a Dynamic Process.
Review of Economics and Statistics, 84 (2).
Duranton, G. and Puga, D. (2001). Nursery Cities: Urban diversity, process innovation, and the life
cycle of products. The American Economic Review, 91 (5).
Duranton, G. and Puga, D. (2004). Microfoundations of Urban Agglomeration Economies. The
Handbook of Regional and Urban Economics, 4.
Duranton, G. and Puga, D. (2005). From sectoral to functional urban specialisation. Journal of
Urban Economics, 57 (2).
Ellison, G. and Glaeser, E. (1997). Geographic Concentration in U.S. Manufacturing Industries: A
Dartboard Approach. Journal of Political Economy, 105 (5).
38
Fujita, M. and Thisse, J.-F. (2002). Economics of Agglomeration: Cities Industrial Location, and
Regional Growth. Cambridge.
Gallant, R. and Nychka, D. (1987). Semi-Nonparametric Maximum Likelihood Estimation. Econometrica, 55(2), 363–390.
Gallant, R. and Tauchen, G. (1989). Semi-Nonparametric Estimation of Conditionally Constrained
Heterogeneous Processes: Asset Pricing Applications. Econometrica, 57(5), 1091–1120.
Gourieroux, C. and Monfort, A. (1993). Simulation Based Inference. Journal of Econometrics, 59,
5–33.
Hansen, L. P. (1982). Large Sample Properties of Generalized Method of Moments Estimators.
Econometrica, 50 (4), 1029–1053.
Henderson, V., Kunkoro, A., and Turner, M. (1995). Industrial Development in Cities. Journal of
Political Economy, 103 (5).
Holmes, T. J. (2010). The Diffusion of Wal-Mart and Economies of Density. Econometrica.
Holmes, T. J. and Stevens, J. J. (2002). Geographic Concentration and Establishment Scale. Review
of Economics and Statistics, 84 (4).
Hopenhayn, H. (1992). Entry, Exit, and Firm Dynamics in Long Run Equilibrium. Econometrica,
60 (5).
Hopenhayn, H. A. and Rogerson, R. (1993). Job Turnover and Policy Evaluation: A General
Equilibrium Analysis. The Journal of Political Economy, 101 (5).
Hotelling, H. (1929). Stability in Competition. The Economic Journal, 39 (153).
Jacobs, J. (1969). The Economy of Cities. Random House.
Krugman, P. (1991). Increasing Returns and Economic Geography. Journal of Political Economy,
99 (3).
Lucas, R. E. and Rossi-Hansberg, E. (2002). On the Internal Structure of Cities. Econometrica,
70 (4).
Marshall, A. (1890). Principles of Economics. Macmillian.
Melitz, M. J. (2003). The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry
Productivity. Econometrica, 71 (6).
39
Mills, E. (1967). An Aggregative Model of Resource Allocation in a Metropolitan Area. American
Economic Review, 57, 197–210.
Muth, R. (1969). Cities and Housing. University of Chicago Press. Chicago.
Rosenthal, S. S. and Strange, W. C. (2001). The Determinants of Agglomeration. The Journal of
Urban Economics, 50 (2).
Rosenthal, S. S. and Strange, W. C. (2003). Geography, Industrial Organization, and Agglomeration.
Review of Economics and Statistics, 85 (2).
Rosenthal, S. S. and Strange, W. C. (2004). Evidence on the Nature and Sources of Agglomeration
Economies. The Handbook of Regional and Urban Economics, 4.
Rossi-Hansberg, E. (2004). Optimal Urban Land Use and Zoning. Review of Economic Dynamics,
7.
Rossi-Hansberg, E. and Wright, M. L. J. (2007). Establishment Size Dynamics in the Aggregate
Economy. American Economic Review, 97 (5).
40